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{{Short description|Black holes appearing from quantum spacetime fluctuations}} {{technical|date=December 2020}}In quantum gravity, a virtual black holeJOURNAL, Virtual Black Holes and Space–Time Structure | SpringerLink, Foundations of Physics, October 2018, 48, 10, 1134–1149, 10.1007/s10701-017-0133-0, 't Hooft, Gerard, 189842716, free, is a hypothetical micro black hole that exists temporarily as a result of a quantum fluctuation of spacetime.S. W. Hawking (1995) "Virtual Black Holes" It is an example of quantum foam and is the gravitational analog of the virtual electron–positron pairs found in quantum electrodynamics. Theoretical arguments suggest that virtual black holes should have mass on the order of the Planck mass, lifetime around the Planck time, and occur with a number density of approximately one per Planck volume.Fred C. Adams, Gordon L. Kane, Manasse Mbonye, and Malcolm J. Perry (2001), "Proton Decay, Black Holes, and Large Extra Dimensions", Intern. J. Mod. Phys. A, 16, 2399.The emergence of virtual black holes at the Planck scale is a consequence of the uncertainty relation A.P. Klimets. (2023). Quantum Gravity. Current Research in Statistics & Mathematics, 2(1), 141-155. where R_{mu} is the radius of curvature of spacetime small domain, x_{mu} is the coordinate of the small domain, ell_{P} is the Planck length, hbar is the reduced Planck constant, G is the Newtonian constant of gravitation, and c is the speed of light. These uncertainty relations are another form of Heisenberg's uncertainty principle at the Planck scale.{| class="toccolours collapsible collapsed" width="60%" style="text-align:left"!Proof |Indeed, these uncertainty relations can be obtained on the basis of Einstein's equations{{Equation box 1|indent=:|equation=G_{munu} + Lambda g_{munu} = {8 pi G over c^4} T_{munu}|cellpadding|border|border colour = #0073CF|background colour=#F5FFFA}}where G_{munu} = R_{munu} - {R over 2} g_{munu} is the Einstein tensor, which combines the Ricci tensor, the scalar curvature and the metric tensor; Lambda is the cosmological constant; а T_{munu} is the energy-momentum tensor of matter; pi is the mathematical constant pi; c is the speed of light; and G is the Newtonian constant of gravitation.Einstein suggested that physical space is Riemannian, i.e. curved and therefore put Riemannian geometry at the basis of the theory of gravity. A small region of Riemannian space is close to flat space.P.A.M. Dirac(1975), General Theory of Relativity, Wiley Interscience, p.9 For any tensor field N_{munu...}, we may call N_{munu...}sqrt{-g} a tensor density, where g is the determinant of the metric tensor g_{munu}. The integral int N_{munu...}sqrt{-g},d^4x is a tensor if the domain of integration is small. It is not a tensor if the domain of integration is not small, because it then consists of a sum of tensors located at different points and it does not transform in any simple way under a transformation of coordinates.P.A.M. Dirac(1975), General Theory of Relativity, Wiley Interscience, p.37 Here we consider only small domains. This is also true for the integration over the three-dimensional hypersurface S^{nu}.Thus, the Einstein field equations for a small spacetime domain can be integrated by the three-dimensional hypersurface S^{nu}. HaveKlimets A.P., Philosophy Documentation Center, Western University-Canada, 2017, pp.25–32Since integrable space-time domain is small, we obtain the tensor equation{{Equation box 1|indent=:|equation=R_{mu}=frac{2G}{c^3}P_{mu}|cellpadding|border|border colour = #0073CF|background colour=#F5FFFA}}where P_{mu}=frac{1}{c}int T_{munu}sqrt{-g},dS^{nu} is the component of the 4-momentum of matter, R_{mu}=frac{1}{4pi}intleft (G_{munu} + Lambda g_{munu}right )sqrt{-g},dS^{nu} is the component of the radius of curvature small domain.The resulting tensor equation can be rewritten in another form. Since P_{mu}=mc,U_{mu} then where r_s is the Schwarzschild radius, U_{mu} is the 4-speed, m is the gravitational mass. This record reveals the physical meaning of the R_{mu} values as components of the gravitational radius r_text{s}.In a small area of space-time is almost flat and this equation can be written in the operator form or{{Equation box 1|indent=:|title=The basic equation of quantum gravity

Psi(x_{mu})rangle=hat R_{mu}|Psi(x_{mu})rangle|cellpadding|border|border colour = #0073CF|background colour=#F5FFFA}}Then the commutator of operators hat R_{mu} and hat x_{mu} is [hat R_{mu},hat x_{mu}]=-2iell^2_{P} From here follow the specified uncertainty relations{{Equation box 1|indent=:|equation=Delta R_{mu}Delta x_{mu}geell^2_{P}|cellpadding|border|border colour = #0073CF|background colour=#F5FFFA}}Substituting the values of R_{mu}=frac{2G}{c^3}m,c,U_{mu} and ell^2_{P}=frac{hbar,G}{c^3}and reducing identical constants from two sides, we get Heisenberg's uncertainty principle Delta P_{mu}Delta x_{mu}=Delta (mc,U_{mu})Delta x_{mu}gefrac{hbar}{2} In the particular case of a static spherically symmetric field and static distribution of matter U_{0}=1, U_i=0 ,(i=1,2,3) and have remained Delta R_{0}Delta x_{0}=Delta r_text{s}Delta rgeell^2_{P} where r_text{s} is the Schwarzschild radius, r is the radial coordinate. Here R_0=r_text{s} and x_0=c,t=r, since the matter moves with velocity of light in the Planck scale.Last uncertainty relation allows make us some estimates of the equations of general relativity at the Planck scale. For example, the equation for the invariant interval dS в in the Schwarzschild solution has the form dS^2=left( 1-frac{r_text{s}}{r}right)c^2dt^2-frac{dr^2}{ 1-{r_text{s}}/{r}}-r^2(dOmega^2+sin^2Omega dvarphi^2) Substitute according to the uncertainty relations r_text{s}approxell^2_P/r. We obtain dS^2approxleft( 1-frac{ell^2_{P}}{r^2}right)c^2dt^2-frac{dr^2}{ 1-{ell^2_{P}}/{r^2}}-r^2(dOmega^2+sin^2Omega dvarphi^2) It is seen that at the Planck scale r=ell_P space-time metric is bounded below by the Planck length (division by zero appears), and on this scale, there are real and virtual Planckian black holes.Similar estimates can be made in other equations of general relativity. For example, analysis of the Hamilton–Jacobi equation for a centrally symmetric gravitational field in spaces of different dimensions (with help of the resulting uncertainty relation) indicates a preference (energy profitability) for three-dimensional space for the emergence of virtual black holes (quantum foam, the basis of the "fabric" of the Universe.). This may have predetermined the three-dimensionality of the observed space.Prescribed above uncertainty relation valid for strong gravitational fields, as in any sufficiently small domain of a strong field space-time is essentially flat.

If virtual black holes exist, they provide a mechanism for proton decay.JOURNAL, Dangerous implications of a minimum length in quantum gravity, 2008, 10.1088/0264-9381/25/19/195013, 0803.0749, Bambi, Cosimo, Freese, Katherine, Classical and Quantum Gravity, 25, 19, 195013, 2008CQGra..25s5013B, 2027.42/64158, 2040645, This is because when a black hole's mass increases via mass falling into the hole, and is theorized to decrease when Hawking radiation is emitted from the hole, the elementary particles emitted are, in general, not the same as those that fell in. Therefore, if two of a proton's constituent quarks fall into a virtual black hole, it is possible for an antiquark and a lepton to emerge, thus violating conservation of baryon number.JOURNAL, Proton decay and the quantum structure of space–time, 2019, 10.1139/cjp-2018-0423, 1903.02940, Al-Modlej, Abeer, Alsaleh, Salwa, Alshal, Hassan, Ali, Ahmed Farag, Canadian Journal of Physics, 97, 12, 1317–1322, 2019CaJPh..97.1317A, 1807/96892, 119507878, The existence of virtual black holes aggravates the black hole information loss paradox, as any physical process may potentially be disrupted by interaction with a virtual black hole.The black hole information paradox, Steven B. Giddings, arXiv:hep-th/9508151v1.

See also

• Quantum foam
• Virtual particle
• Quantum tunnelling

References

{{reflist}}{{black holes}}{{quantum-stub}}